Right-preordered groups from a categorical perspective

TL;DR

This paper explores the categorical structure of right-preordered groups, detailing limits, colimits, and exactness, and reveals their similarities with monoids and properties of semidirect products.

Contribution

It provides an explicit categorical description of right-preordered groups, including limits, colimits, and split extensions, highlighting their algebraic and categorical properties.

Findings

Right-preordered groups share properties with monoids.

Semidirect products of ordered groups naturally admit right-preorders.

The category exhibits specific limits, colimits, and exactness properties.

Abstract

We study the categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, and studying some exactness properties. We show that, from an algebraic point of view, the category of right-preordered groups shares several properties with the one of monoids. Moreover, we describe split extensions of right-preordered groups, showing in particular that semidirect products of ordered groups have always a natural right-preorder.